The goal of this activity is to analyze a linear relationship with data gathered in context. Students examine data collected by submerging equal-size objects in a graduated cylinder that is partially filled with water, and then measuring the resulting volume by observing the level of the water (MP2).

This activity uses a 100 ml graduated cylinder filled with an initial amount of 60 ml of water and number cubes whose volume is approximately 3.7 ml each. Depending on the materials you gather, your measurements may be different. Be sure to leave enough space so many number cubes can be added before the water reaches the top.

There are three versions of this activity, two for a shorter 20-minute time frame and one for a longer 40-minute time frame. For the shorter versions, either the teacher performs a demonstration adding number cubes to the cylinder, and the whole class works with this data, or students work in groups with the digital applet. For the longer version, students gather their own data.

Note that for either of the non-digital versions, because of measurement error, the data may not lie exactly on a line, and different slope triangles may lead to slightly different values for the slope. Monitor for students who get slightly different values, and invite them to share during the discussion.

In the digital version of the activity, students use an applet to simulate dropping marbles into a graduated cylinder full of water. The applet allows students to set the starting water level, drop marbles into the water one by one, and reset the simulation as needed. This activity works best when each group has access to the applet. The digital version is beneficial because it requires less time, less setup and cleanup than physical manipulatives, and it allows students to focus on the linear relationship rather than taking precise measurements. If students don't have individual access, displaying the applet for all to see would be helpful during the Launch.

Display the equation for all to see, or if a different amount of water or objects with a different volume was used, display the equation corresponding to the class data. Explain that this equation represents the situation they saw today. Discuss:

• “What do the variables represent?” ( is the total volume in the cylinder in ml and is the number of objects added to the cylinder.)

• “Where do you see a rate of change in this equation? What does it mean in this situation?” (The rate of change is 3.7 and is seen as the coefficient of the variable . It is the volume of each object in ml or the increase of volume in the cylinder in ml each time an object is dropped in. )

• “What does the number 60 represent?” (60 is the initial amount of water in the cylinder in ml.)

Emphasize that the equation can be interpreted as saying:

If necessary, discuss the precision of the computed slope. It makes sense that every object increases the volume by the same amount (because the number cubes, for example, are all the same size). For smaller objects or graduated cylinders with larger diameters, it may be difficult to accurately measure the change for 1 object. If students add objects in larger sets, the slope calculation will tell the amount of change for one object.

The purpose of this activity is for students to consider how the parts of an equation are related to the linear context it describes. Without a graph, students must identify what the slope and vertical intercept of an equation mean.

The goal of this discussion is for students to connect parts of an equation to the context it describes, without a graph. Invite as many students as time allows to share the stories they created for the second question. Then discuss:

• “How does the 5 show up in each story? Do they have anything in common?” (Answers vary.)

• “How does the show up in each story? Do they have anything in common?” (Answers vary.)